| Summary: | We present the results of a systematic study of the evolution of low- and intermediate-mass X-ray binaries (LMXBs and IMXBs). Using a standard Henyey-type stellar-evolution code and a standard model for binary interactions, we have calculated 100 binary evolution sequences containing a neutron star and a normal-type companion star, where the initial mass of the secondary ranges from 0.6 to 7\Ms and the initial orbital period from $\sim 4 $hr to $\sim 100 $d. This grid of models samples the entire range of parameters one is likely to encounter for LMXBs and IMXBs. The sequences show an enormous variety of evolutionary histories and outcomes, where different mass-transfer mechanisms dominate in different phases. Very few sequences resemble the classical evolution of cataclysmic variables, where the evolution is driven by magnetic braking and gravitational radiation alone. Many systems experience a phase of mass transfer on a thermal timescale and may briefly become detached immediately after this phase (for the more massive secondaries). In agreement with previous results (\markcite{Tauris1}Tauris and Savonije 1999), we find that all sequences with (sub-)giant donors up to $\sim 2\Ms$ are stable against dynamical mass transfer. Sequences where the secondary has a radiative envelope are stable against dynamical mass transfer for initial masses up to $\sim 4\Ms$. (abridged)
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